the water depth in a harbor rises and falls over time. the function ( f(t)=4.1sinleft(\frac{pi}{6}t…

the water depth in a harbor rises and falls over time. the function ( f(t)=4.1sinleft(\frac{pi}{6}t - \frac{pi}{3}\right)+19.7 ) models the water depth, in feet, after ( t ) hours. during the first 24 hours, at what times does the water depth reach a maximum? at 5 and 17 hours at 11 and 23 hours at 2, 8, 14, and 20 hours at 5, 11, 17, and 23 hours

the water depth in a harbor rises and falls over time. the function ( f(t)=4.1sinleft(\frac{pi}{6}t - \frac{pi}{3}\right)+19.7 ) models the water depth, in feet, after ( t ) hours. during the first 24 hours, at what times does the water depth reach a maximum? at 5 and 17 hours at 11 and 23 hours at 2, 8, 14, and 20 hours at 5, 11, 17, and 23 hours

Answer

Explanation:

Step1: Find the general solution for maximum

The maximum of (y = A\sin(Bt - C)+D) occurs when (Bt - C=\frac{\pi}{2}+ 2k\pi), (k\in\mathbb{Z}). For (f(t)=4.1\sin(\frac{\pi}{6}t-\frac{\pi}{3}) + 19.7), set (\frac{\pi}{6}t-\frac{\pi}{3}=\frac{\pi}{2}+2k\pi).

Step2: Solve for (t)

First, simplify the equation (\frac{\pi}{6}t-\frac{\pi}{3}=\frac{\pi}{2}+2k\pi). Add (\frac{\pi}{3}) to both sides: (\frac{\pi}{6}t=\frac{\pi}{2}+\frac{\pi}{3}+2k\pi=\frac{3\pi + 2\pi}{6}+2k\pi=\frac{5\pi}{6}+2k\pi). Multiply both sides by (\frac{6}{\pi}): (t = 5+12k).

Step3: Find (t) values in (0\leq t\leq24)

When (k = 0), (t=5). When (k = 1), (t=5 + 12=17). When (k = 2), (t=5+24 = 29>24) (rejected). When (k=- 1), (t=5-12=-7<0) (rejected).

Answer:

at 5 and 17 hours